Abstract
Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Some mathematical proofs are formal proofs. A formal proof consists in a sequence of formulas in a formal language, each of which is either an axiom or the result of applying one of the explicitly stated rules of inference to previous formulas in the sequence. However, this paper is about gaps in ordinary mathematical proofs and ordinary mathematical proofs, as they occur in mathematical practice, are not formal proofs.
In a literature review on non-deductive methods in mathematics, Baker (2015) has a section on gaps in proofs where he recounts Fallis’ work in (2003). He notes that the notion of ‘proof gap’ is in need of further clarification (Baker 2015, Sect. 2.1.2). Hamami (2014) takes steps in this direction by providing a more detailed account than Fallis of basic mathematical inferences. In particular, he applies Dag Prawitz’s account of valid inference (e.g., Prawitz 2012) to mathematics. Hamami suggests that we think of Fallis’ categories of gaps relative to the resulting account of valid mathematical inference. For our purposes, it is not necessary that we have a clear notion of what a complete proof is. An intuitive notion of this is enough.
We thank a referee for suggesting that we address this question.
We thank the two referees for pressing us to be more clear on the relationship between enthymematic gaps, untraversed gaps, and universally untraversed gaps.
We are grateful to a referee for pressing us to develop further the philosophical motivations and consequences of the empirical study.
The number of female tenured mathematicians that are Danish and working at Danish universities is very small.
We thank Henrik Kragh Sørensen and Mikkel Willum Johansen for suggesting this approach.
Mathematicians may on average be less thorough when they are not acting as referees, since the interviews suggest that referees take the task of determining whether the submitted paper is sound very seriously (Andersen 2017).
Easwaran (2009) offers an account of why mathematicians are generally unwilling to accept probabilistic proofs, but do accept proofs that skip steps and are long and complicated. (A probabilistic proof does not deductively establish its conclusion but establishes that there is some, often specifiable, high probability of the conclusion being true.) In a footnote, he states, referring to enthymematic gaps, that he thinks that “the sorts of proof gaps that are acceptable are the ones that relevant experts can see and still be convinced” by the proof (Easwaran 2009, p. 355). My interviews support this picture.
At the same time, a referee may very well ask an author to provide less detail—i.e. to leave more enthymematic gaps—in the straightforward parts of a proof. We are grateful to a referee for pressing us to clarify our claim here.
This is a very partial response to the following remarks by Grice: “Finicky over-elaboration of intervening steps is frowned upon, and in extreme cases runs the risk of forfeiting the title of reasoning. In speech, such over-elaboration would offend against conversational maxims, against (presumably) some suitably formulated maxim of Quantity. In thought, it will be branded as pedantry or neurotic caution. At first sight, perhaps, one would have been inclined to say that greater rather than lesser explicitness the better. But now it looks as if proper explicitness is an Aristotelian mean, and it would be good some time to enquire what determines where that mean lies” (Grice 2001, p. 16; quoted in Paseau 2016, p. 187).
References
Andersen, L. E. (2017). On the nature and role of peer review in mathematics. Accountability in Research,24, 177–192.
Azzouni, J. (2004). The derivation-indicator view of mathematical practice. Philosophia Mathematica,12, 81–105.
Azzouni, J. (2009). Why do informal proofs conform to formal norms? Foundations of Science,14, 9–26.
Baker, A. (2015). Non-deductive methods in mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. https://plato.stanford.edu/archives/fall2015/entries/mathematics-nondeductive/. Accessed 14 December 2016.
Davis, P. J. (1972). Fidelity in mathematical discourse: Is one and one really two? The American Mathematical Monthly,79, 252–263.
Devlin, K. (2003). When is a proof? In Devlin’s angle. A column written for the Mathematical Association of America. https://www.maa.org/external_archive/devlin/devlin_06_03.html. Accessed 14 December 2016.
Dutilh Novaes, C. (2016). Reductio ad absurdum from a dialogical perspective. Philosophical Studies,173, 2605–2628.
Dutilh Novaes, C. (2017). A dialogical conception of explanation in mathematical proofs (forthcoming).
Easwaran, K. (2009). Probabilistic proofs and transferability. Philosophia Mathematica,17, 341–362.
Ernest, P. (1994). The dialogical nature of mathematics. In P. Ernest (Ed.), Mathematics, education and philosophy: An international perspective (pp. 33–48). London: Falmer Press.
Fallis, D. (2003). Intentional gaps in mathematical proofs. Synthese,134, 45–69.
Geist, C., Löwe, B., & Van Kerkhove, B. (2010). Peer review and knowledge by testimony in mathematics. In B. Löwe & T. Müller (Eds.), PhiMSAMP. Philosophy of mathematics: Sociological aspects and mathematical practice (pp. 155–178). London: College Publications.
Grcar, J. (2013). Errors and corrections in mathematics literature. Notices of the American Mathematical Society,60, 418–425.
Grice, H. P. (2001). Aspects of reason. Oxford: Oxford University Press.
Hamami, Y. (2014). Mathematical rigor, proof gap and the validity of mathematical inference. Philosophia Scientiæ,18, 7–26.
Hardwig, J. (1991). The role of trust in knowledge. Journal of Philosophy,88, 693–708.
Johansen, M. W., & Misfeldt, M. (2016). An empirical approach to the mathematical values of problem choice and argumentation. In B. Larvor (Ed.), Mathematical cultures: The London meetings 2012–2014 (pp. 259–269). Basel: Birkhäuser.
Müller-Hill, E. (2009). Formalizability and knowledge ascriptions in mathematical practice. Philosophia Scientiæ,13, 21–43.
Müller-Hill, E. (2011). Die epistemische Rolle formalisierbarer mathematischer Beweise. Formalisierbarkeitsorientierte Konzeptionen mathematischen Wissens und mathematischer Rechtfertigung innerhalb einer sozio-empirisch informierten Erkenntnistheorie der Mathematik. Inaugural-Dissertation. Bonn: Rheinische Friedrich-Wilhelms-Universität. http://hss.ulb.uni-bonn.de/2011/2526/2526.htm. Accessed 14 December 2016.
Paseau, A. C. (2011). Mathematical instrumentalism, Gödel’s theorem, and inductive evidence. Studies in History and Philosophy of Science,42, 140–149.
Paseau, A. C. (2016). What’s the point of complete rigour? Mind,125, 177–207.
Pelc, A. (2009). Why do we believe theorems? Philosophia Mathematica,17, 84–94.
Prawitz, D. (2012). The epistemic significance of valid inference. Synthese,187, 887–898.
Rav, Y. (1999). Why do we prove theorems? Philosophia Mathematica,7, 5–41.
Rav, Y. (2007). A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica,15, 291–320.
Tanswell, F. (2015). A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica,23, 295–310.
Thurston, W. P. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society,30, 161–177.
Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education,39, 431–459.
Acknowledgements
Part of the research for this paper was conducted while I was a postdoc at the Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Brussels, Belgium. Two years ago, I had the pleasure of interviewing eight mathematicians about their refereeing practices. I thank them for their time and openness. The paper has benefited greatly from comments from Henrik Kragh Sørensen, Karen Francois, Mikkel Willum Johansen, and two anonymous referees. I also thank Henrik and Mikkel for their help in developing the interview questions. Earlier versions of the paper were presented at the 2016 Society for Philosophy of Science in Practice conference (Glassboro, New Jersey), the 2016 Novembertagung on the history and philosophy of mathematics (Sønderborg, Denmark), the workshop ‘Mathematical evidence and argument: Historical, philosophical, and educational perspectives’ (Bremen, Germany, 2017), the workshop ‘Group knowledge and mathematical collaboration’ workshop (Oxford, UK, 2017), and the 2017 Nordic Network for Philosophy of Science meeting (Copenhagen, Denmark). I would like to thank the audiences for useful comments.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that she has no conflict of interest.
Rights and permissions
About this article
Cite this article
Andersen, L.E. Acceptable gaps in mathematical proofs. Synthese 197, 233–247 (2020). https://doi.org/10.1007/s11229-018-1778-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-018-1778-8